I’m trying to find the best framework for dealing with the following problem. With ordinal Y there is no link function that works the majority of times even within a specific subject matter field. I often compare goodness of fit for a proportional odds ordinal logistic model with a log-log link ordinal model (proportional hazards) because they are so different. I sense that it’s best to allow “link function uncertainty”, but I don’t seek a final result that mixes the two links but instead one that results in a single effect measure I can communicate to the researcher, e.g., I’d like to get a posterior distribution for a certain odds ratio or hazards ratio, and be able to communicate exactly which one this is.

Is there a way to do this that will reflect in the final posterior distribution the uncertainty in choosing that model? Are there other ways to think about this?

As a side note I don’t typically use the probit ordinal model because of the lack of an easy interpretation for the model parameters.